Classifications

• • G—PHYSICS
  • G02—OPTICS
  • G02B—OPTICAL ELEMENTS, SYSTEMS OR APPARATUS
  • G02B3/00—Simple or compound lenses
  • G02B3/0087—Simple or compound lenses with index gradient

Definitions

• the present invention relates in general to lens elements. It relates in particular to a lens element made from an optically transparent lens blank, all or part of which has a gradient refractive index.
• incorporating one or more unconventional refractive optical elements in a multi-element lens or optical system may improve performance, reduce element count, or both.
• Optical design codes capable of dealing with gradient index materials, at any level, have not, until relatively recently, been commercially available.
• Gradient index materials were originally prepared by applying surface treatments to homogeneous materials. Such treatments include chemical immersion, ion implantation and the like. Such methods of preparation are generally not capable of providing a wide range of index values in a material, or providing a material which is actually thick enough to make a practical lens blank. It is widely believed that GRIN optical elements would be most effective if the refractive index gradient of the varied radially, i.e., in a direction perpendicular to the element's optical axis. An axial gradient material, in which refractive index varies with distance along an element's optical axis, is generally believed not to offer any benefit that could not be achieved by an aspheric element.
• GRIN elements are generally discussed as part of an optical system including one or more homogenous elements.
• the GRIN property in such cases is considered as providing no more than an additional degree of freedom for an optical designer.
• System optimization often involves merely a trial-and-error approach, which, unfortunately, is encouraged by the power, speed and availability of modern computers. Because of this, it is often difficult to determine precisely the exact contribution of a GRIN element to the system performance. It is probable that, by considering GRIN elements in this way, the full potential of GRIN elements is not being realized.
• a spherical refractive optical element has all common optical aberrations, to some degree, depending on its actual shape and material.
• spherical elements are usually Gauss system providing low coma, and several forms of doublet groups for providing low chromatic distortion.
• a GRIN material could be formulated that would allow a single lens element to be produced in a wide variety of shapes while still having at least one selected aberration be essentially zero, or, if not zero, at least some substantially constant value. It would be particularly useful if such a GRIN material could be formulated from an axial GRIN material made by the above referenced technique of Hagerty et al..
• a GRIN element in a system of optical elements, to correct a selected system aberration, without perturbing another system aberration which had been already adequately corrected.
• a GRIN element could be as standard an optical building-block as any of the well-known conventional multi-element groups are at present.
• a gradient refractive index lens element comprises first and second surfaces formed on respectively first and second sections of a transparent monolithic unit.
• the first and second surfaces are generally arranged on a common optical axis.
• a selected one of the first and second sections has a gradient refractive index varying in a direction parallel to the optical axis.
• the other section has a substantially homogeneous refractive index.
• the gradient and homogeneous refractive indices have substantially the same value at any location on a junction between the first and second sections.
• the lens element has a shape-factor, X, defined by an equation
• Rl is the radius of curvature of the first surface and R2 is the radius of curvature of the second surface.
• the gradient refractive index may be selected such that at least one third order aberration selected from the group consisting of spherical, coma, and astigmatism is substantially zero for at least one value of the shape-factor.
• first and second surfaces are formed on an optically transparent material having a gradient rsfractive index.
• the first and second surfaces are generally arranged on a common optical axis.
• the gradient refractive index varies according to a predetermined non-linear function of distance measured in a direction parallel to the optical axis.
• the non-linear function may be a cubic or a quadratic function varying only in an axial direction.
• the function may be a linear or non ⁇ linear function, varying in a radial as well as an axial direction.
• the index gradient function may be selected such that at least one third order aberration selected from the group consisting of spherical, coma , and astigmatism may be either zero or some substantially constant value over a range of shape-factors, for a fixed value of index change.
• the gradient refractive index function is a cubic function selected such that third order spherical aberration is substantially eliminated for positive shape-factors between about 0.5 and 2.0 for an index-change value of about -0.2.
• first and second surfaces are formed on an optically transparent material having a gradient refractive index.
• the first and second surfaces are generally arranged on pivot-point.
• a pivot-point for a selected aberration may be created at any shape-factor in the range between about -2.0 and 2.0.
• first and second surfaces are formed on an optically transparent material having a gradient optical dispersion.
• the first and second surfaces are generally arranged on a common optical axis.
• the gradient dispersion material has a dispersion gradient characterized by a dispersion varying in a direction parallel to the optical axis.
• the shape-factor of the lens element provides a first axial chromatic aberration for red and blue light passing through the lens element.
• the dispersion gradient provides a second axial chromatic aberration for red and blue light passing through the lens element.
• the shape-factor and the dispersion gradient may be selected such that the first and second chromatic aberrations for red and blue light are equal in magnitude to each other, but opposite in sign, thereby correcting chromatic aberration and providing an achromatic lens element.
• This chromatic aberration correction mechanism is independent of the base refractive index of the transparent optically inhomogeneous material.
• the base refractive index may itself be inhomogeneous. Because of this, the possibility exists that a gradient index material having appropriate dispersion gradient properties may be used to provide an element which is DESCRIPTION OF THE DRAWINGS.
• FIG. 1 is an on-axis axial cross section view illustrating a bi-convex lens formed from a linear axial gradient refractive index material.
• FIG. IA is a graph schematically illustrating a linear refractive index profile of the lens element of FIG. 1.
• FIG. 2 is a graph schematically illustrating third order spherical aberration for linear axial gradient singlet lens elements as a function of shape-factor and index-change.
• FIG. 3 is a graph schematically illustrating one range of allowed cubic and quadratic gradient index profiles for singlet lens elements in accordance with the present invention.
• FIG. 4 is a graph schematically illustrating another range of allowed cubic and quadratic gradient
• FIG. 5 is a graph schematically illustrating third order spherical aberration as a function of shape-factor and index-change for quadratic axial gradient, singlet lens elements in accordance with the present invention.
• FIG. 6 is a graph schematically illustrating third order spherical aberration as a function of shape-factor and index-change for cubic axial gradient, singlet lens elements in accordance with the present invention.
• FIGS 7A-C are graphs schematically illustrating third order coma as a function of stop position, shape- factor, and index-change for linear axial gradient, singlet lens elements.
• FIGS 8A-C are graphs schematically illustrating 'third order coma as a function of stop position, shape- factor, and index-change for quadratic axial gradient, singlet lens elements in accordance with the present invention.
• FIGS 9A-C are graphs schematically illustrating third order astigmatism as a function of stop position, shape-factor, and index-change for linear axial gradient, singlet lens elements in accordance with the present invention.
• FIGS 10A-C are graphs schematically illustrating third order astigmatism as a function of stop position.
• FIG. 11 is an on-axis axial cross-section view illustrating a bi-convex lens formed from a monolithic unit including a graded refractive index material and an optically homogenous material in accordance with the present invention.
• FIG. 11A is a graph schematically illustrating a refractive index profile of the lens element of FIG. 11.
• FIG. 11B is an enlarged portion of the refractive index profile of FIG. 11A schematically illustrating refractive index continuity at a junction between the graded index and homogeneous materials of FIG. 11.
• FIG. 11C is an on-axis axial cross-section view illustrating a positive meniscus lens element formed from a monolithic unit including a graded refractive index material and an optically homogenous material in accordance with the present invention.
• FIG. 12 is a graph schematically illustrating third order spherical aberration as a function of shape-factor for GRIN cap elements in accordance with the present invention.
• FIG. 13 is an on-axis axial cross-section view illustrating a bi-convex lens formed from a spherical gradient refractive index material in accordance with the present invention. gradient refractive index material in accordance with the present invention.
• FIGS 15A-B are graphs schematically illustrating third order spherical aberration as a function of shape- factor, index-change and iso-index plane curvature for, piano gradient, singlet lens elements in accordance with the present invention.
• FIG. 16 schematically illustrates an achromatic, gradient dispersion, singlet lens element in accordance with the present invention.
• FIG. 17A-B are graphs schematically illustrating transverse ray aberration plots at three wavelengths for the achromatic singlet lens element of FIG. 16.
• FIG. 18A-B are graphs schematically illustrating transverse ray aberration plots for a prior art homogeneous lens element having the same shape as the gradient dispersion lens element of FIG. 16.
• FIG. 19 is a graph schematically illustrating transverse ray aberration as a function of wavelength for the achromatic singlet lens element of FIG. 16.
• FIG. 20 is a graph schematically illustrating transverse ray aberration as a function of wavelength for a prior art homogeneous lens element having the same
• FIG. 21 is a graph schematically illustrating chromatic aberration versus shape-factor for gradient dispersion singlet lens elements in accordance with the present invention.
• n noo + n 0 ⁇ z + n 02 z 2 + no 3 z 3 +...
• n 0 o, n 02r n 03 are respectively linear, quadratic, and cubic coefficients, defining an index profile in a direction parallel to the optical axis.
• the distance z is assumed to be measured in the direction in which light is incident on the lens element when it is oriented for its intended use.
• surfaces of the element are designated first and second surfaces, counted in the direction of incident light.
• a curved surface is described as having a positive radius of curvature, or positive curvature, when it appears convex when viewed in the direction of incident light, and negative when it appears concave when viewed in the direction of incident light.
• ⁇ is the total amount of a given aberration present in an optical system.
• S is the contribution from surface i.
• S is a non-linear function of factors such as the curvature of the surface, refractive indices on either side of the surface, and axial and chief ray intercept coordinates (in an x-y reference plane normal to the optical axis, or z direction).
• A is the contribution from surface asphericity, if any.
• spherical surfaces are considered, although the use of spherical surfaces in conjunction with the present invention is certainly not precluded. In spherical surfaces A is assumed to be zero.
• G is a surface contribution introduced by the refractive index gradient, and, in general, is a non ⁇ linear function of factors such as the gradient index type and profile, and axial and chief ray intercept coordinates.
• T is the gradient index contribution, and is due to propagation of light through the gradient index medium. It is usually referred to as the transfer term, and is an extremely non-linear function of factors such as the gradient index type and profile, and axial and chief ray intercept coordinates.
• FIG. 1 a biconvex element 20 having first and second surfaces, 22 and 24 respectively, arranged on a common optical axis 26, is illustrated.
• Surface 22 according to the above discussed convention, has a positive curvature and surface 24 has a negative curvature.
• Surfaces 22 and 24 are formed on an optically continuous, graded refractive index material 28 having a linear refractive index gradient.
• the refractive index profile of material 28 is represented graphically in FIG. IA.
• refractive index decreases in the z direction from a maximum value (n h ), at point 30 on surface 22, to a minimum value (nl), at a point 32 on surface 24.
• Phantom line 34 indicates the outline of an original blank or block of material from which element 20 is formed.
• ⁇ n is positive and is defined by an equation.
• the surface contribution of the refractive index increases along the surface in a direction away from point 32, and has a highest value at point 38.
• marginal rays are bent less strongly than axial rays.
• marginal rays are bent more strongly than axial rays.
• Rl is the radius of curvature of the first surface
• R2 is the radius of curvature of the second surface
• Rl and R2 When Rl and R2 are negative, X has a value less than -1.0. When Rl is piano (infinite radius of curvature) and R2 is negative, X has a value of -1.0. When Rl is positive, and R2 is negative, but less in absolute value (magnitude) than Rl, X has a value of between -1.0 and 0.0. When Rl is positive, and R2 is negative and equal in magnitude to Rl, X has a value of 0.0.
• X has a value between 0.0 spherical aberration (SA3) versus shape-factor and ⁇ n for F/3 singlet lens elements having a thickness of 15.0 millimeters (mm), and a diameter of 80.0 mm.
• SA3 0.0 spherical aberration
• Curves of FIG. 2 represent the performance when RMS spot size for the lens elements is optimized for a point monochromatic light source located at infinity.
• Variables in the computation were the values of Rl and R2, and the slope (+ or -) of the index gradient. Constraints were ⁇ n and the high extreme index value of an axially graded glass blank from which the lens elements would be made. This was fixed at a value of 1.95. The low index value was dictated by whatever ⁇ n was assumed to be.
• curve HI for a homogeneous lens shows optimum performance at a shape-factor of 1.0.
• Computations indicate that in order to correct third order spherical aberrations using a linearly graded index, ⁇ n must be less than or equal to -.03 (as indicated approximately by curves G1C) or greater than or equal to 0.25. Provided this condition is satisfied, there is one shape-factor for any given ⁇ n which will have zero spherical aberration.
• ⁇ n for which a range of values of shape-factor may be substantially corrected, even if aberration at all a pivot-point. The existence of such a point may seem without detailed analysis to be surprising. It indicates that at shape-factors close to or equal to zero, no amount of axial linear gradient is effective in correcting third order spherical aberration.
• Equation (7) indicates that the mechanism which creates the pivot-point is an effective cancellation of spherical aberration produced by the first surface of an element, by spherical aberration produced by the second surface of that element.
• the discovery of the pivot-point provides useful insight into designing an optical system incorporating one or more elements having a linear axial refractive index gradient. It also points the way to designing gradient index materials, having an index profile other than linear, from which elements having certain unique and desirable properties may be fabricated. This will be discussed in detail below.
• the pivot-point PI indicates make most effective use of a linear gradient.
• this is accomplished by providing an axial gradient index material in which refractive index varies axially in accordance with a predetermined non-linear function of distance measured in the direction of the optical axis.
• Providing a non-linear gradient allows index gradient slope, as well as the actual index value, to vary axially.
• a non-linear function may be selected such effect of first and second surfaces.
• Preferred index gradient functions are quadratic and cubic functions, i.e., functions in which coefficients 11 02 and n 0 3 of equation (2) have a finite value.
• index gradient slope of profile QC3 was lowest at the first surface of the element and highest at the second surface of the element.
• index gradient slope of profile QC4 was lowest at the first surface of the element, and highest at the second surface of the element.
• a gradient index profile could not have an inflection point between the maximum and minimum values of refractive index. Additionally, all points on the profile were required to have a higher value than points on a linear profile (L3 of FIG. 3 and L4 of FIG. 4) joining the maximum and minimum values of refractive index n h and ni, except, of course, at the actual maximum and minimum values.
• FIGS 5 and 6 graphically illustrate computed third order spherical aberration versus shape- factor and ⁇ n for F/3 singlet lens elements having a thickness of 15.0 millimeters (mm), and a diameter of 80.0 mm.
• FIGS 5 and 6 It is evident from FIGS 5 and 6, that providing a quadratic or a cubic profile is not as effective as might be expected in eliminating the pivot-point at shape-factors close to 0.0. Further, it can be seen that a quadratic profile does not provide a curve which changes the sign of the third order spherical aberration at any ⁇ n between -0.4 and 0.4. Significant, however, is that a quadratic profile having a ⁇ n value of about - 0.05 (FIG 5, curve G2) provides substantially corrected elements at all shape-factors between about -.05 and - 2.0. A cubic profile having a ⁇ n value of about -.20 will provide precise correction (about zero spherical aberration) for elements having any shape-factor between about .05 and 2.0.
• the shape-factor of the element is desired to be 1.5. Changing the shape-factor of a linear gradient element to this value would cause a substantial and undesirable increase in third order spherical aberration. The same shape change in an element having a quadratic gradient, however, would produce a negligible change in third order spherical aberration.
• FIGS 7A- C which graphically depict third order tangential coma (TC03), as a function of shape-factor and ⁇ n, for linear gradient F/3 singlet lens elements having a thickness of 15.0 millimeters (mm), a diameter of 80.0 mm, and a stop gradient.
• TC03 third order tangential coma
• FIGS 9A-C illustrate the third order sagittal astigmatism (SAS) performance of elements having the same dimensional specification as the lens of FIGS 7A-C and having a linear axial gradient.
• FIGS 10A-C illustrate the third order sagittal astigmatism performance of elements having the same dimensional specification as the lens of FIGS 7A-C, and having a quadratic axial gradient.
• the coma pivot-point may exist at positive and negative shape-factors between about -0.5 and 0.5, depending on placement of the stop. This is indicated, for example, by arrow P2 in FIGS 7A-C. Comparing FIGS 7A-C and FIGS 8A-C, it can be seen that coma curves for a range of value of ⁇ n are more tightly grouped for quadratic than for linear index profiles. invariant for ⁇ n values between about -.2 and .4 over a range of shape-factors between about -2.0 and 2. This is an example of an iso-aberrant lens in which the aberration is not zero or substantially corrected, but has a substantially constant, non-zero value.
• FIGS 11, and 11A-C another method of providing an iso-aberrant range of lens elements using a graded index material is illustrated.
• a biconvex element 40 (see FIG. 11) having first and second surfaces, 42 and 44 respectively, arranged on common optical axis 26, is illustrated.
• Unit 46 comprises sections 46A and 46B, which are aligned in the direction of optical axis 26.
• Section 46A has an axially graded refractive index
• section 46B has an essentially homogeneous refractive index.
• Surface 42 is formed on section 46A and surface 44 is formed on section 46B.
• Section 46A may be described as forming a gradient refractive index cap for element 50. Accordingly, in the following description, monolithic elements of the type exemplified by element 50 are referred to as "GRIN-cap" elements.
• section 46A has a linear refractive index profile, wherein refractive index decreases in the z direction (negative ⁇ n) from a maximum value (n h ), at point 50 on surface 42, to a minimum value (ni), at a junction h ⁇ w ⁇ ⁇ o t.nn?. dfi ⁇ and 4fiB inriir-atprt hv broken line
• unit 46 is fabricated as a single optically continuous block of material, for example, according to the above-referenced fusion technique of Hagerty et al..
• junction 52 In a material fabricated in this way, junction 52 would not be sharp and singular, as depicted at point 54 of FIG 11A, but would be diffuse. It would thus provide a graded, optically continuous refractive index transition, as indicated by line 56 of FIG. 11B. This would effectively eliminate any possibility of a reflective interface occurring at junction 52. A reflective interface would need to be taken into account in optical design calculations, and, further, may adversely effect properties of element 40.
• element 50 by bonding together separately-fabricated graded and homogeneous sections is not precluded. In order for element 50 to function as a single element, however, index difference between the sections, or any intervening adhesive material, should not exceed (I NEED A NUMBER TO SUPPORT THE TERM SUBSTANTIALLY EQUAL).
• section 46 is illustrated in FIG. 11 as comprising only that portion of element on which surface 42 is formed, this should not be construed as a limitation in GRIN-cap elements. Rather, formation of a GRIN-cap element requires only that one surface be formed entirely on a graded refractive index material and the other surface be formed entirely on a material having an essentially homogeneous refractive index. This effectively a designer as a variable, for example, to take advantage of transfer contribution of the GRIN material. Clearly, however using only sufficient GRIN material to form a surface is particularly desirable as the cost of GRIN material, while not prohibitive for many applications, is, presently, greater than many common optical glasses.
• FIG. 12 illustrates computed third order spherical aberration as function of shape-factor for a range of such elements having shape- factors between about -2.0 and 2.0. Optimization for each element was carried out in accordance with above- discussed criteria. For elements having shape-factors of 0.0 or greater, it was assumed that first surface 42 was generated on axially graded refractive index material having negative index-change (- ⁇ n), and second surface 44 was generated on essentially homogeneous material. For elements having shape-factors less than 0.0, (see FIG.
• axial GRIN elements may be shaped to form a pivot-point at which a particular aberration is essentially constant for a wide range of both positive and negative ⁇ n values.
• the value of such a pivot-point element together with a limitation presented by having such a pivot-point close to zero shape-factor, is discussed above.
• spherical aberration even non-linear axial GRIN elements appear to provide only limited ability for selection of pivot-point position. For coma and elements having positive values of shape-factor than negative values of shape.
• FIGS 13 and 14 illustrate two methods of providing such a radial component.
• a lens element 60 has first and second surfaces 62 and 64 respectively arranged on a common optical axis 26.
• Element 60 is formed from a spherical gradient refractive index material 66.
• Refractive index at any point within element 60 is a predetermined function of distance from a point 68 on optical axis 26. Refractive index is thus constant at any point on an imaginary spherical plane having a center of curvature at point 28. Such a plane may be conveniently described as an "iso-index" plane.
• the effect of such a spherical gradient is to create a material which not only has a refractive index that is actually graded, but also varies in a direction perpendicular to the axis, for example in direction y.
• the refractive index has both axial and radial gradient components.
• FIG. 14 is illustrated a method by which benefits of a spherical gradient may be realized without significant difficulties in manufacture or analysis.
• a lens element 80 having first and second surfaces 82 and 84 respectively, is formed from a gradient refractive index material 86.
• Material 86 is formed such that refractive index varies according to a predetermined function along optical axis 26, and is constant, at any point within the element, on an imaginary spherical plane having a fixed predetermined radius of curvature, and having a center of curvature on optical axis 26.
• imaginary planes 91, 92, and 93 all have a fixed radius of curvature R4, and have centers of curvature 94, 95 and 96 respectively extending along optical axis 26.
• Such a material may be termed a piano gradient material, or a shell gradient material
• a material such as material 86 may be readily fabricated by a variation of the method of Hagerty. By curvature of the iso-index planes makes lenses formed from a piano gradient material easier to analyze than a spherical gradient material.
• FIGS 15A-B show computed third order spherical aberration as a function of shape-factor and index- change ⁇ n for 80 mm diameter F/3 lens elements having a thickness of about 15 mm. Optimization was conducted as described above for other examples of GRIN lens elements. In each case, the lens element was assumed to be formed from a piano gradient refractive index material having a refractive index increasing linearly along on the optical axis.
• iso-index planes (e.g. 91-93 of FIG. 14) were assumed to have a radius of curvature of 50 mm. It can be seen that the piano gradient element may be corrected at shape-factors between about -0.5 and 0.5, which is not the case for elements having only an axially graded refractive index.
• the corrections can be effected by selecting an appropriate value of ⁇ n between about .05 and .1. This correction is possible, at least in part, because the pivot-point P4 occurs at a shape- factor of about 0.75. This correction process appears to favor elements having negative values of shape-factor.
• iso-index planes were assumed to have radius of curvature of 25 mm.
• correction for shape-factors between -0.5 and 0.5 is possible when ⁇ n has a value of about .025. This is realized, at least in part, by a displacement of the pivot-point P5 to a shape-factor of about 1.6. however, should not be construed as restrictive.
• a piano gradient lens element may have many useful combinations of positive or negative curvature of iso index planes, positive or negative values of ⁇ n, and linear or non-linear variation of refractive index along the optical axis. In particular, such combinations provide a means by which a GRIN lens element may be selected to have a pivot-point at any shape-factor between about -2.0 and 2.0.
• Dispersion is a property of all optical glasses. It is the variation of refractive index with wavelength (color) of light.
• dispersion of an optical material is specified by a dimensionless number known as the Abbe number.
• the Abbe number or "V” number is inversely proportional to difference in refractive index of a material at the "F” and “C” emission lines of hydrogen.
• the F and C lines have wavelengths of 486.13 nanometers (nm) and 656.27 nm respectively.
• Axial primary chromatic aberration of a lens is For a thick lens element of a given focal length, and composed of homogeneous glass, axial chromatic aberration is the sum of first and second surface contributions to the chromatic aberration. Each surface contribution depends, in turn, on factors including the dispersion of the homogeneous glass, the refractive index change across the surface, and the angle of incidence of a marginal ray going from the axial object point to the edge of the lens element aperture.
• a lens formed from a gradient refractive index material includes both surface contributions and a transfer contribution to the aberration. Such is also the case for chromatic aberration. It is generally believed, however, that such a transfer contribution is negligible in an axial gradient material, as an axial gradient material does not provide any refractive power.
• achromatic singlet lens in accordance with the present invention being optically continuous.
• Axial gradient GRIN materials of the type discussed above may be considered, for some computation purposos, as a stack of infinitesimally thin plates, each sequentially higher or lower in refractive index (or dispersion) than the other. Plane parallel plates in a converging beam of light will indeed cause chromatic aberration, and, further, in a sense opposite that aberration caused by surfaces converging the beam. Increasing or decreasing dispersion of the plates through the stack increases the aberration effect of the plates.
• v 0 ⁇ is the initial dispersion, on axis, of the gradient dispersion material
• ⁇ is the axial change in dispersion
• x is the lens element shape-factor as defined above.
• ⁇ is a measure of the dispersion of the GRIN material which is directly linearly proportional to the difference in refractive index of the material at the hydrogen F and C lines (WE NEED TO STATE THE CONSTANT OF PROPORTIONALITY !).
• Equation (8) indicates that chromatic aberration depends on change in dispersion in the material, but not on refractive index of the material. It is also clear from the equation that a small initial dispersion, combined with a large change in dispersion, results in small values of shape-factor in an achromatic singlet element. The equation also indicated that positive change in dispersion is preferably used to correct elements having a negative shape-factor, and negative change in dispersion is preferably used to correct lens elements having a positive shape-factor.(OR IS IT LESS THAN OR GREATER THAN 1.011)
• equation (8) is sufficiently accurate to define basic conditions for providing an achromatic singlet element, the value of X provided by the equation is only approximate.
• the value refinement is now commonplace in the art to which the present invention pertains, and, indeed, was used to compute specifications and performance of achromatic GRIN singlet elements described below.
• Element 98 has first and second surfaces 100 and 102 arranged on a common optical axis 26.
• Element 98 is formed from an axially graded refractive index material having a dispersion increasing in a direction parallel to optical axis 26.
• the refractive gradient here, is selected only to provide a desired dispersion gradient.
• the refractive index gradient neither augments nor compromises the color correction process. It will be evident from descriptions given above, however, that the refractive gradient may affect one or more third order aberrations.
• the focal length is 100.0 mm and the diameter of the element is 5.0 mm.
• Initial dispersion v 0 ⁇ is -0.0384; change in dispersion ⁇ is -0.069; and lens shape-factor is 2.18.
• Transverse ray aberration plots for element 98, at three wavelengths including the F and C lines, are depicted in FIGS 17A-B. It can be seen that curves for the F and C lines are coincident, indicating an absence element 98 is depicted in FIG. 19. Equal aberration values at the hydrogen F and C wavelengths indicate absence of axial chromatic aberration. By way of comparison, transverse ray aberration versus wavelength for a homogeneous lens element having the same shape as element 98 is depicted in FIG. 20. Here, substantial chromatic aberration is evident.
• axial refractive index gradient lens elements include iso-aberrant elements in which at least one third order aberration is either zero or some substantially constant value at shape- factors between -2.0 and 2.0. Also included are pivot- point elements which are arranged such that a selected third order aberration is invariant with refractive index change. Non-linear and compound refractive index gradients are arranged such that only a relatively small index-change in an element is required to provide desired iso-aberrant properties. A dispersion gradient material is used to provide achromatic singlet lens elements.
• Refractive index suitable for fabricating such elements may be formulated using a proven layer fusion method. Refractive index changes required for elements in accordance with the present well as index dispersion values if this is necessary to satisfy system correction requirements.

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• 1994
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